Now, to study the second term in the equation (REF ), one needs an explicit formulation of the velocity field \(\mathbf {v}.\)  It has been proved in [1]}, [2]} that the velocity field associated to \((QGSW)_{\lambda }\)  equations writes in the context of vortex patches as an integral on the boundary, namely
\(\mathbf {v}(t,z)=\frac{1}{2\pi }\int _{\partial D_{t}}K_{0}(\lambda |z-\xi |)d\xi ,\) 
